However, it has been observed that \(\Delta _M\)  actually encodes all of the geometry of \(M\)  when viewed as a quadratic operator in the sense that \(\Delta _M\)  obeys a certain product rule [1]}.
This observation suggests that graph Laplacians \(L_h\)  can, in principle, be used directly to give information about \(M\)  previously obtained indirectly from multi-stage algorithms.
For example in recent years, the global topology, or shape, of \(M\)  has often been inferred by thickening the sample points \(X_1,\dots ,X_n\)  into small balls and studying the actual shape of the union of those balls [2]}.
In contrast, that same data can be computed more directly in terms of \(L_h\)  [1]}.
For another example, the Riemannian metric or infinitesimal lengths of paths through the the sample points \(X_1,\dots ,X_n\)  can be estimated through various algorithms [4]}, [1]}, [6]} (some of which even directly use \(\hat{\Delta }_h\)  [1]}.)
Intrinsic manifold distances might then be estimated as shortest path lengths, where path lengths are computed in the sample network in terms of the estimated Riemannian metrics.
It is not clear whether such a hypothetical estimator would be consistent.
In contrast, let \({\mathrm {dist}}_{L;r;\epsilon ;\tau }(a,b)\)  be the following estimate of \(\mathrm {dist}_M(X_a,X_b)\) , the optimal value of the following non-linear optimization problem defined in terms of tuning parameters \(r,\epsilon ,\tau >0\)  and a matrix estimate \(L\)  of \(\Delta _M\)  having the first \(r\)  negative eigenvalues \(\lambda _{1;L}\ge \lambda _{2;L}\cdots \ge \lambda _{r;L}\) ; here \(v_{(i);L}\)  denotes the projection of a vector \(v\in \mathbb {R}^n\)  onto the span of the first \(i\)  eigenspaces of \(L\) , \(\Vert v\Vert _\infty =\max _i|v_i|\) , and \(\odot \)  denotes the coordinate-wise product for vectors:
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